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0901 Submissions

A Revision to Gödel's Incompleteness Theorem by Neutrosophy

According to Smarandache's neutrosophy, the Gödel's incompleteness theorem contains the truth,
the falsehood, and the indeterminacy of a statement under consideration. It is shown in this
paper that the proof of Gödel's incompleteness theorem is faulty, because all possible
situations are not considered (such as the situation where from some axioms wrong results can
be deducted, for example, from the axiom of choice the paradox of the doubling ball theorem
can be deducted; and many kinds of indeterminate situations, for example, a proposition can
be proved in 9999 cases, and only in 1 case it can be neither proved, nor disproved). With
all possible situations being considered with Smarandache's neutrosophy, the Gödel's
Incompleteness theorem is revised into the incompleteness axiom: Any proposition in any
formal mathematical axiom system will represent, respectively, the truth (T), the falsehood (F),
and the indeterminacy (I) of the statement under consideration, where T, I, F are standard or
non-standard real subsets of ]-0, 1+[ . With all possible situations being considered, any
possible paradox is no longer a paradox. Finally several famous paradoxes in history, as
well as the so-called unified theory, ultimate theory and so on are discussed.
Category:Number Theory

N = pi + (N-pi) = p+ (N-p). If p is congruent to N modulo pi, Then (N-p) is a composite integer,
When i = 1, 2,..., r, if p and N are incongruent modulo pi, Then p and (N-p) are solutions of Goldbach's
Conjecture (A); By Chinese Remainder Theorem we can calculate the primes and solutions of Goldbach's
Conjecture (A) with different system of congruence; The (N-p) must have solution of Goldbach's
Conjecture (A), The number of solutions of Goldbach's Conjecture (A) is increasing as N → ∞, and finding
unknown particulars for Hardy-Littewood's Conjecture (A).
Category:Number Theory

On Dark Energy, Weyl Geometry and Brans-Dicke-Jordan Scalar Field

We review firstly why Weyl's Geometry, within the context of
Friedman-Lemaitre-Robertson-Walker cosmological models, can account for both the
origins and the value of the observed vacuum energy density (dark energy).
The source of dark energy is just the dilaton-like Jordan-Brans-Dicke
scalar field that is required to implement Weyl invariance of the
most simple of all possible actions. A nonvanishing value of the vacuum
energy density of the order of 10-123M4Planck is derived in agreement
with the experimental observations. Next, a Jordan-Brans-Dicke gravity
model within the context of ordinary Riemannian geometry, yields also
the observed vacuum energy density (cosmological constant) to very high
precision. One finds that the temporal flow of the scalar field φ(t) in
ordinary Riemannian geometry, from t = 0 to t = to, has the same numerical
effects (as far as the vacuum energy density is concerned) as if
there were Weyl scalings from the field configuration φ(t), to the constant
field configuration φo, in Weyl geometry. Hence, Weyl scalings in Weyl
geometry can recapture the flow of time which is consistent with Segal's
Conformal Cosmology, in such a fashion that an expanding universe may
be visualized as Weyl scalings of a static universe. The main novel result
of this work is that one is able to reproduce the observed vacuum energy
density to such a degree of precision 10-123M4Planck, while still having a
Big-Bang singularity at t = 0 when the vacuum energy density blows up.
This temporal flow of the vacuum energy density, from very high values
in the past, to very small values today, is not a numerical coincidence but
is the signal of an underlying Weyl geometry (conformal invariance) operating
in cosmology, combined with the dynamics of a Brans-Dicke-Jordan
scalar field.
Category:Relativity and Cosmology